# Find the Value Of K For Which Each of the Following System of Equations Has Infinitely Many Solutions : 2x + 3y = K (K − 1)X + (K + 2)Y = 3k - Mathematics

Sum

Find the value of k for which each of the following system of equations has infinitely many solutions :

2x +3y = k

(k - 1)x + (k + 2)y = 3k

#### Solution

The given system of the equation may be written as

2x +3y = k = 0

(k - 1)x + (k + 2)y = 3k = 0

The system of equation is of the form

a_1x + b_1y + c_1 = 0

a_2x + b_2y + c_2 = 0

where a_1 = 2, b_1 = 3, c_1= -k

And a_2 = k -1,b_2 = k + 2, c_2 = 3k

For a unique solution, we must have

a_1/a_2 - b_1/b_2 = c_1/c_2

=> 2/(k-1) = 3/(k +1) = (-k)/(-3k)

=> 2/(k -1) = 3/(k +1) and 3/(k +1) = (-k)/(-3k)

=> 2(k + 2) = 3(k - 1) and 3 xx 3 = k + 2

=> 2k + 4 = 3k - 3 and  9 = k + 2

=> 4 + 3 = 3k - 2k  and 9 - 2 = k

=> 7 = k and 7 = k

k = 7 satisfies both the conditions

Hence, the given system of equations will have infinitely many solutions if  k = 7

Is there an error in this question or solution?

#### APPEARS IN

RD Sharma Class 10 Maths
Chapter 3 Pair of Linear Equations in Two Variables
Exercise 3.5 | Q 19 | Page 73